Display object textured regardless of viewport display mode
As you use the unwrapped plane with image texture (or image imported with Images As Planes addon) to show as the reference, it will lose effects from the Solid shading mode because the plane is a mesh and will behave exactly the same as the one you're modeling. So while changing shading mode to Wireframe no image will be visible. 

Use background images instead; they don't depend on shading mode. Note that they will appear only in preset orthographic views, however those should be used for precision modeling anyway.
It's also possible to use an  empty of Image type (Shift+A > Empty > Image) which can be useful as reference image. It doesn't depend neither on shading modes nor on ortho views.